You must use ProQuake, Qrack, DirectQ or DarkPlaces to play Quake over the internet with modern equipment.
You are encountering the "NAT" problem. You might be able to type "status" in the console after you get the "connection accepted" problem and force it to continue to connect, but you are using a router or have a DSL or Cable modem with a built-in router and ancient old WinQuake isn't completing the handshake.
These clients don't have that issue and are actively maintained and bug fixed:
ProQuake:
http://quakeone.com/proquake (WinQuake like .exe is wqpro443.exe)
Qrack:
http://quakeone.com/qrackDirectQ:
http://mhquake.blogspot.comDarkPlaces:
http://icculus.org/twilight/darkplaces/